A Unified Representation of Density-Power-Based Divergences Reducible to M-Estimation

Abstract: Density-power-based divergences are known to provide robust inference procedures against outliers, and their extensions have been widely studied. A characteristic of successful divergences is that the estimation problem can be reduced to M-estimation. In this paper, we define a norm-based Bregman density power divergence (NB-DPD) -- density-power-based divergence with functional flexibility within the framework of Bregman divergences that can be reduced to M-estimation. We show that, by specifying the function $\phi_\gamma$, NB-DPD reduces to well-known divergences, such as the density power divergence and the $\gamma$-divergence. Furthermore, by examining the combinations of functions $\phi_\gamma$ corresponding to existing divergences, we show that a new divergence connecting these existing divergences can be derived. Finally, we show that the redescending property, one of the key indicators of robustness, holds only for the $\gamma$-divergence.